[EM] weighted pairwise method and compressing-ranks method

[James Green-Armytage]

Chris,

	Today I've been trying to think very seriously about the
compressing-rankings method which you proposed
http://lists.electorama.com/pipermail/election-methods-electorama.com/2003-September/010899.html
	You were quite right when you said that eliminating non-Schwartz set
members helps quite a lot. I think that I hadn't taken that into account
when I read the proposal earlier on this year.
	Note: I would also suggest modifying the proposal such that the ranks are
not compressed unless there is no candidate not beaten by a majority.
(That is, I think that if there is a candidate who is beaten pairwise, but
only by a minority, and every other candidate is beaten by a majority, it
helps strategy-wise to elect that minority-beaten candidate without
further adjustments to the rankings.)
	Anyway, the method does have a certain elegance to it. I think that it is
a very interesting method. 
	I guess that the purpose of this thread is to talk about which method is
preferable, between the compressed ranks method and the weighted pairwise
method which I proposed recently. 
http://fc.antioch.edu/~jarmyta@antioch-college.edu/voting_methods/weighted_pairwise.htm
	Or at least, if it can't be stated with confidence that one method is
preferable to the other, to understand some of their relative strengths
and weaknesses. But to be fair, I'll admit that the weighted pairwise
method is kind of my baby right now, so forgive me if I'm a bit partisan
in my analysis.
	For anyone just tuning in, these are two different Condorcet methods
which use ratings to break majority rule cycles, and they are BLOODY
INTERESTING, I'll have you know ; )

	Let me begin by noting that while the two methods have a great deal in
common, they are different, and not only in a superficial way but in a
fundamental way. Here is an example that shows how they can give different
results:

24: A 100 > B 1 > C 0
24: A 100 > C 1 > B 0
22: B 100 > C 99 > A 0
4: B 100 > C 1 > A 0
1: B 100 > A 1 > C 0
22: C 100 > B 1 > A 0
3: C 100 > A 1 > B 0

	You may or may not have noticed that I am more or less recycling the same
old Bush Dean Kerry example that I have been using all week, except with
some minor adjustments. By the way, this example is a big fat tie in
regular winning votes Condorcet, sorry about that, but it won't be a
problem here.

	My method identifies the pairwise defeats and gives them weighted
magnitudes as follows
A-->B: 4779
B-->C: 542
C-->A: 4679
	The B-->C defeat is weakest and is therefore dropped, and C (formerly
known as John Kerry) wins.
	(Thanks again to Brian Olson for encoding this method into his ratings
calculator.)

	Let me note here that given a three candidate Schwartz set, your
compressing ranks method is essentially equivalent to Condorcet completed
by approval voting. That is, if my preferences were originally A>>B>C,
they will become A>B=C in the cycle-breaking part of the procedure,
equivalent to an approval ballot. This is intuitive enough because I
initially indicated three levels of preference, and the cycle-breaking
method is to reduce the number of levels by one, leaving me with two
levels of preference.

	Hence, given the above examples, we can express the compressed ballots in
the cycle breaking round as approval ballots.
24: AB
24: A
22: BC
4: B
1: B
22: C
3: C
	And I think that the approval scores of the candidates are sufficient to
tell us the winner.
A: 48
B: 49
C: 47
	So, where my method chooses C, your method chooses B. 
	The margin of victory for C in my method is rather large; the strength of
the B-->C defeat doesn't come close to the other defeats in weighted
magnitude. The margin of B's victory in your method is rather small, but
an example can be constructed where the margin is comfortable in both
methods. For example, if some of the ACB voters rate C above 50, they will
bolster B's approval score in your method without changing the result in
mine.

	Well, I guess that's all I've really figured out so far: your method is
equivalent to an approval completion in a 3 candidate top cycle. I haven't
done any examples with 4 or 5 candidate cycles yet, I don't know how those
would turn out.
	But anyway, to me completing Condorcet with approval voting is not
necessarily the best thing. I don't know. I mean, it's not the worst thing
by any means. Again, it has a certain elegant simplicity to it. But I
guess that you can make some of the same criticisms of this method that
you can make of an ordinary approval completion, except perhaps the
criticisms are a bit more toned down.
	There's the cooperation/defection dilemma, for one. The above example
kind of illustrates it, in a way. That is, there were two B>C>A voters who
didn't rank C above 50, but all of the C>B>A voters ranked B above 50. If
those two B>C>A voters had ranked C above 50, then C would have won
instead of B. So you could imagine that the renegade B>C>A gained an
advantage by defecting. Of course, if a few B voters defect, and a few C
voters defect, then A will win.
	So, it's the old dilemma, although it seems quite a bit more distant than
in regular approval. For one thing, this dilemma only comes to play in the
event of a cycle. But then, I'd point out that your whole method (and mine
as well) only comes to play in the event of a cycle. So that criticism may
not be as distant as it may seem. However, again, I don't know what the
deal would be in a top cycle with more candidates.
	I don't know what else I can say about an approval completion of
Condorcet. Again, I don't think that it's the worst possibility, but I'm
not sure that it's the best either.
	I guess I'm going to go on preferring the weighted pairwise method to the
compressing-ranks method unless someone convinces me otherwise, or unless
I get some new idea about it myself.
	Sorry, maybe that's about all I can think to say right now, even though I
worked on it for several hours this afternoon.

my best,
James Green-Armytage

P.S. Again, I seem to remember reading some sort of failure example that
you wrote for the compressing-ranks method. Was that just a dream or a
hallucination? It ended with the sentence "which is too bad, but it still
might be the best single-winner method." Or was that someone else, talking
about something else? Sorry, I can't find it.